Objectives
In this section you will learn how to:
express curves and surfaces with parametric equations
Section6.2Parametric Curves: \(f\colon {\mathbb{R}}\to {\mathbb{R}}^m \)
ΒΆExercise6.2.1
A horse runs around an elliptical track. Its position at time \(t\) is given by the function \(\vec r(t)=(2\cos t, 3\sin t).\) We could alternatively write this as \(x=2\cos t, y=3\sin t\text{.}\)
(a)
What are \(n\) and \(m\) when we write this function in the form \(\vec r\colon {\mathbb{R}}^n\to {\mathbb{R}}^m\text{?}\)
(b)
Construct a graph of this function.
(c)
Next to a few points on your graph, include the time \(t\) at which the horse is at this point on the graph. Include an arrow for the horse's direction.
(d)
How many dimensions do you need to graph this function?
(e)
If we wanted to plot time (t) on its own axis, how many dimensions would we need?
Use the SageMathCell below, or this Wolfram Alpha link to plot the functions.
Notice in the problem above that we placed a vector symbol above the function name, as in \(\vec r\colon {\mathbb{R}}^n\to {\mathbb{R}}^m\text{.}\) When the target space (codomain) is 2-dimensional or larger, we place a vector above the function name to remind us that the output is more than just a number.
In the next problem, we keep the input as just a single number \(t\text{,}\) but the output is now a vector in \(\mathbb{R}^3\text{.}\)
Exercise6.2.2
A jet begins spiraling upwards to gain height. The position of the jet after \(t\) seconds is modeled by the equation \(\vec r_2(t)=(2\cos t, 2\sin t, t).\) We could alternatively write this as \(x=2\cos t,\, y=2\sin t,\, z=t\text{.}\)
(a)
What are \(n\) and \(m\) when we write this function in the form \(\vec r\colon {\mathbb{R}}^n\to {\mathbb{R}}^m\text{?}\)
(b)
Construct a graph of this function by picking several values of \(t\) and plotting the points resulting from \((2\cos t, 2\sin t, t)\text{.}\)
(c)
Next to a few points on your graph, include the time \(t\) at which the jet is at this point on the graph. Include an arrow for the jet's direction.
(d)
How many dimensions do you need to graph this function?
Use the SageMathCell below, or this Wolfram Alpha link to plot the functions.
Exercise6.2.3
On a separate piece of paper (you'll be expanding this later) create a table with columns for:
problem number
function
\(n\)
\(m\)
number of dimensions required to graph
Go back over the previous problems in this unit and fill in the table.